I need to prove that Aristotle's Angle Unboundedness Axiom holds in hyperbolic geometry and I don't really know where to start. The problem says that we can take a segment parallel to one of the legs ...

I have read numerous paper over area calculation in hyperbolic geometry but just can't seem to understand how to calculate a triangle's area in hyperbolic geometry. It would be nice to have a proof ...

Given a regular or uniform tessellation of hyperbolic plane, is there a way to find a group of cells that will tile the whole plane?
For example: in the $(6,6,7)$ tessellation (truncated $\{3,7\}$), ...

I see in the Wikipedia article on the hyperboloid model and also in this other Math.SE question about the hyperboloid model that this is how you calculate distance on the hyperboloid model:
Let $u = ...

Can there be a segment on a hyperbolic plane that goes from point $(0,0)$ to $(0,0)$ in the hyperbolic plane. There are some rules, though for this to work:
1) The segment must apply to the rules of ...

Let $M=\mathbb{H}/SL(2,\mathbb{Z})$ be the modular surface (which is noncompact but finite volume with the volume induced by the constant negative curvature metric inherited from $\mathbb{H}$). Any ...

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for ...

At the moment it is to hot for real mathematics but I wanted to have a function that relates the lengths of the real sides of an Ideal Lambert quadrilateral
An Ideal Lambert quadrilateral (my term, ...

I had a space-ship wreck in an unknown world of some kind of moths. I could observe geometer moths working. Everything looked strange. The moths claimed that they were using only straight edges and ...

There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but ...

In [Zeghib: Laminations et hypersurfaces géodésiques des variétés hyperboliques, Annales scientifiques de l'ENS, 1991] it is shown, that in a compact manifold of negative curvature, there exists only ...

Let $M$ be a compact Riemannian manifold, $\bar M \to M$ be its universal covering and $\phi \in Isom(M)$ be an isometry of $M$. Is it true that, if $\phi$ is isotopic to the identity map of $M$, than ...

In Is the shortest path in flat hyperbolic space straight relative to Euclidean space?
I answered by refering to the Triangle Inequality
(https://en.wikipedia.org/wiki/Triangle_inequality , Euclid's ...

I'm interested in the structure of homeo- and diffeomorphism groups of oriented surfaces, especially in hyperbolic case. For example, does the homeomorphism group retracts on the diffeomorphism group ...

I'm reading some notes on hyperbolic surfaces by François Labourie and there's an exercise I can't figure out.
I have to prove that the length l(c) of a curve does not depend on the subdivision. It's ...

I am confused by this picture:
https://en.wikipedia.org/wiki/File:HyperboloidProjection.png
What is wrong with projecting from the origin, and using the disc at $t = 1$? After doing some computation ...

Given the Euclidean coordinates of two points (p1, p2) and (q1, q2) in the unit circle, how do I construct the Euclidean circle x^2 + y^2 + fx + gy+1 representing the hyperbolic d-line on the poincare ...

I'm trying to find some intuition in Lorentz transformations. I understand that they are basically rotations by imaginary angle of vector of the form $\{ict,x\}$ (for $1+1$ space-time dimensions), and ...

I'm looking for a proof that in the Poincare disk model the circumference of a circle of radius $r$ is $2\pi \sinh r$.
I have seen this result in many places but I haven't been able to find a proof. ...

I'm reading Stillwell's Geometry of Surfaces but I'm having a little bit of trouble because my background in calculus isn't great, I'm struggling with these problems:
In the upper half-plane model, ...